Mathematical methods in chemistry and physics, michael e starzak

www.elsolucionario.net www.elsolucionario.net Mathetnatical Methods in Chetnistry and Physics www.elsolucionario.net Mathematical Methods in Chemistry and Physics Michael E Starzak State University of New York at Binghamton Binghamton, New York Springer Science+Business Media, LLC www.elsolucionario.net Library of Congress Cataloging in Publication Data Starzak, Michael E Mathematical methods in chemistry and physics / Michael E Starzak p cm Includes bibliographical references and index ISBN 978-1-4899-2084-3 ISBN 978-1-4899-2082-9 (eBook) DOI 10.1007/978-1-4899-2082-9 Chemistry - Mathematics Physics - Mathematics I Title QD39.3.M3S73 1989 510'.2454-dcI9 88-32133 CIP © 1989 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1989 Softcover reprint of the hardcover 1st edition 1989 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher www.elsolucionario.net Preface Mathematics is the language of the physical sciences and is essential for a clear understanding of fundamental scientific concepts The fortuna te fact that the same mathematical ideas appear in a number of distinct scientific areas prompted the format for this book The mathematical framework for matrices and vectors with emphasis on eigenvalue-eigenvector concepts is introduced and applied to a number of distinct scientific areas Each· new application then reinforces the applications which preceded it Most of the physical systems studied involve the eigenvalues and eigenvectors of specific matrices Whenever possible, I have selected systems which are described by x or x matrices Such systems can be solved completely and are used to demonstrate the different methods of solution In addition, these matrices will often yield the same eigenvectors for different physical systems, to provide a sense of the common mathematical basis of all the problems For example, an eigenvector with components (1, -1) might describe the motions of two atoms in a diatomic molecule or the orientations of two atomic orbitals in a molecular orbital The matrices in both cases couple the system components in a parallel manner Because I feel that x 2, x 3, or soluble N x N matrices are the most effective teaching tools, I have not included numerical techniques or computer algorithms A student who develops a clear understanding of the basic physical systems presented in this book can easily extend this knowledge to more complicated systems which may require numericalor computer techniques The book is divided into three sections The first four chapters introduce the mathematics of vectors and matrices In keeping with the book's format, simple examples illustrate the basic concepts Chapter intro duces finite-dimensional vectors and concepts such as orthogonality and linear independence Bra-ket notation is introduced and used almost exclusively in subsequent chapters Chapter introduces function space vectors To illustrate the strong paralleis between such spaces and N-dimensional vector spaces, the concepts of Chapter 1, e.g., orthogonality and linear independence, are developed for function space vectors Chapter introduces matrices, beginning with basic matrix operations and concluding with an introduction to eigenvalues and eigenvectors v www.elsolucionario.net vi PreCace and their properties Chapter introduces practical techniques for the solution of matrix algebra and ca1culus problems These include similarity transforms and projection operators The chapter concludes with some finite difference techniques for determining eigenvalues and eigenvectors for N x N matrices Chapters 5-8 apply the mathematics to the major areas of normal mode analysis, kinetics, statistieal mechanics, and quantum mechanies The examples in the chapter demonstrate the paralleis between the one-dimensional systems often introdu~ed in introductory courses and multidimensional matrix systems For example, the single vibrational frequency of a one-dimensional harmonie oscillator intro duces a vibrating molecule where the vibrational frequencies are related to the eigenvalues of the matrix for the coupled system In each chapter, the eigenvalues and eigenvectors for multieomponent coupled systems are related to familia~ physical concepts The final three chapters introduce more advanced applications of matriees and vectors These include perturbation theory, direct products, and fluctuations The final chapter introduces group theory with an emphasis on the nature of matrices and vectors in this discipline The book grew from a course in matrix methods I developed for juniors, seniors, and graduate students Although the book was originally intended for a one-semester course, it grew as I wrote it The material can still be covered in a one-semester course, but I have arranged the topics so chapters can be eliminated without disturbing the flow of information The material can then be covered at any pace desired This material, with additional numerical and programming techniques for more complicated matrix systems, could provide the basis for a two-semester course Since the book provides numerous examples in diverse areas of chemistry and physics, it can also be used as a supplemental· text for courses in these areas Each chapter concludes with problems to reinforce both the concepts and the basic ex am pies developed in the chapter In all cases, the problems are directed to applications I wish to thank my wife Anndrea and my daughters Jocelyn and Alissa for their support throughout this project and Alissa for converting my pencil sketches into professional line drawings I am grateful to the students whose comments and suggestions aided me in determining the most effective way to present the material I also wish to thank my readers in advance for their suggestions for improvement Michael E Starzak Binghamton, New York www.elsolucionario.net Contents Vectors 1.1 Vectors 1.2 Vector Components 1.3 The Scalar Product 1.4 Scalar Product Applications 1.5 Other Vector Combinations 1.6 Orthogonality and Biorthogonality 1.7 Projection Operators 1.8 Linear Independence and Dependence 1.9 Orthogonalization of Coordinates 1.10 Vector Calculus Problems 1 14 20 26 32 37 40 46 52 Function Spaces 2.1 The Function as a Vector 2.2 Function Scalar Products and Orthogonality 2.3 Linear Independence , 2.4 Orthogonalization of Basis Functions 2.5 Differential Operators 2.6 Generation of Special Functions 2.7 Function Resolution in a Set of Basis Functions 2.8 Fourier Series Problems 55 55 57 63 67 70 77 83 90 98 Matrices 3.1 Vector Rotations 3.2 Special Matrices 3.3 Matrix Equations and Inverses 3.4 Determinants 3.5 Rotation of Co ordinate Systems 3.6 Principal Axes 3.7 Eigenvalues and the Characteristic Polynomial 101 101 109 114 119 125 133 140 vii www.elsolucionario.net viii Contents 3.8 Eigenveetors 3.9 Properties of the Charaeteristic Polynomial 3.10 Alternate Teehniques for Eigenvalue and Eigenveetor Determination Problems 145 152 Similarity Transforms and Projections 4.1 The Similarity Transform 4.2 Simultaneous Diagonalization 4.3 Generalized Charaeteristie Equations 4.4 Matrix Deeomposition Using Eigenveetors 4.5 The Lagrange-Sylvester Formula 4.6 Degenerate Eigenvalues 4.7 Matrix Funetions and Equations 4.8 Diagonalization of Tridiagonal Matriees 4.9 Other Tridiagonal Matrices 4.10 Asymmetrie Tridiagonal Matriees Problems 165 165 171 176 181 186 191 199 205 211 216 221 Vibrations and Normal Modes 5.1 Normal Modes 5.2 Equations of Motion for a Diatomie Moleeule 5.3 Normal Modes for Nontranslating Systems 5.4 Normal Modes Using Projeetion Operators 5.5 Normal Modes for Heteroatomic Systems 5.6 A Homogeneous One-Dimensional Crystal 5.7 Cyclie Boundary Conditions 5.8 Heteroatomie Linear Crystals 5.9 Normal Modes for Moleeules in Two Dimensions Problems 225 225 232 240 246 252 258 264 271 276 286 Kinetics 6.1 Isomerization Reaetions 6.2 Properties of Matrix Solutions of Kinetie Equations 6.3 Kineties with Degenerate Eigenvalues 6.4 The Master Equation 6.5 Symmetrization of the Master Equation 6.6 The Wegseheider Conditions and Cyclic Reaetions 6.7 Graph Theory in Kinetics 6.8 Graphs for Kinetics 6.9 Mean First Passage Times 6.10 Evaluation of Mean First Passage Times 6.11 Stepladder Models Problems 289 289 294 300 309 315 321 332 337 340 346 351 356 157 161 www.elsolucionario.net Contents ix Statistical Mechanics 7.1 The Wind-Tree Model 7.2 Statistical Mechanics of Linear Polymers 7.3 Polymers with Nearest-Neighbor Interactions 7.4 Other One-Dimensional Systems 7.5 Two-Dimensional Systems 7.6 Non-Nearest-Neighbor Interactions 7.7 Reduction of Matrix Order 7.8 The Kinetic Ising Model Problems 359 359 366 373 379 385 389 393 399 407 Quantum Mechanics 8.1 Hybrid Atomic Orbitals 8.2 Matrix Quantum Mechanics 8.3 Hückel Molecular Orbitals for Linear Molecules 8.4 Hückel Theory for Cyclic Moleeules 8.5 Degenerate Molecular Orbitals for Cyclic Moleeules 8.6 The Pauli Spin Matrices 8.7 Lowering and Raising Operators 8.8 Projection Operators Problems 409 409 415 421 430 437 444 452 461 467 Driven Systems and Fluctuations 9.1 Singlet-Singlet Kinetics 9.2 Multilevel Driven Photochemical Systems 9.3 Laser Systems 9.4 lonic Channels 9.5 Equilibrium and Stationary-State Properties 9.6 Fluctuations about Equilibrium 9.7 Fluctuations during Reactions 9.8 The Kinetics of Single Channels in Membranes Problems 469 469 475 482 487 493 500 509 517 524 10 Other Techniques: Perturbation Theory and Direct Products 10.1 Development of Perturbation Theory 10.2 First-Order Perturbation Theory-Eigenvalues 10.3 First-Order Perturbation Theory-Eigenvectors 10.4 Second-Order Perturbation Theory-Eigenvalues 10.5 Second-Order Perturbation Theory-Eigenvectors 10.6 Direct Sums and Products 10.7 A Two-Dimensional Coupled Oscillator System Problems 527 527 532 539 546 550 557 564 571 11 Introduction to Group Theory 573 11.1 Vectors and Symmetry Operations 573 www.elsolucionario.net 637 Section 11.9 • Direct Products of Group Elements o -1O)®C0 C =C ®E=C = CO ~-~ C o 0 x (-1 I=E(8)I= 0) ® CO o -1 o o C o 0"=C ®i= 0 0) ® (-1 -1 ~n 0 0 0)1 j) (11.9.16) ~) D ( 11.9.17) ~) j) (11.9.18) The direct product of each of these diagonal x matrices is also diagonal The irreducible representations for the four-element group appear as the block diagonal elements of these matrices The diagonal elements of the x matrices gave the irreducible representations for the groups Their direct product gave the irreducible representations for the product group Thus, the character table for the larger group is genera ted as a direct product of the character tables of the constituent groups The technique can be illustrated for the direct product of the D group and the Ci group The D group differs from the C group since it contains three C operations about axes which are perpendicular to the z axis for the C operations The symmetry is illustrated in Figure 11.36 Rotation about any of the three axes in the plane will reproduce the vectors The character table for D has characters identical to those for C 3v' i.e., E -1 -1 o (11.9.19) www.elsolucionario.net Chapter 11 • Introduction to Group Theory 638 Figure 11.36 A set of vectors which satisfy the symmetry operations of D • The operations include C and C~ operations about z and three C2 rotations about the three axes in the xy plane The subscripts and define the character of C • The elements are listed by class, i.e., two C operations and three C operations The character table for Ci IS E (11.9.20 ) -1 The direct product of the two tables is 1 1) D ®C i = ( 1 -1 -1 ®(1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 o -2 (11.9.21 ) Although this is the correct character table for the D 3d group, the symmetry elements for each column must be determined Some belong to the original groups Since D occurs first in the direct product, its three classes are coupled with E from Ci and they are repeated in the new table (E, 2C3 , and 3C2 ) The fourth column couples E from D with i from Ci and gives the i symmetry www.elsolucionario.net 639 Section 11.9 • Direct Products of Group Elements Figure 11.37 Combination of the C3 and i operations to produce a new S~ operation element The fifth and sixth columns will contain new symmetry elements generated from the coupling of the C3 and C operations, respectively, with the i operation Since the classes C and C respectively contain two and three elements while the class for i has one element, columns and are classes with two and three elements, respectively The first new symmetry operation results from a direct product of C or C~ and i The three vectors of Figure 11.37 are 120° apart A C rotation, followed by an inversion, places the vector at -60° This is the S~ operation in which the vector is rota ted - 60° and then reflected through a plane perpendicular to the rotation axis (Figure 11.37) The direct product of the C~ and i operations gives the S6 operation (Figure 11.38) Two S6 operations in sequence are equivalent to a C3 operation The final set of three symmetry operations are the direct product of the C operations and the inversion operation These operations are equivalent to a reflection through a plane perpendicular to the xy plane and oriented halfway between the C rotation axes (Figure 11.39) The reflection plane is not present for D and the presence of the inversion element in the group is required for its appearance The direct product functions in two ways with groups The direct product of two symmetry operations from the constituent subgroups generates new symmetry elements for the group The direct product of the two character tables for the constituent subgroups gene rates the full character table for the direct product group When the full character table is known, it can be used to establish the irreducible representations present for a molecule which contains all symmetries tabulated for the group Figure 11.38 Combination of the Ci and i operations to produce the S6 operation www.elsolucionario.net Chapter 11 • Introduction to Group Theory 640 I a a I b C b b o b I b c Figure 11.39 The (f d rellection operation generated as a combination of C and i operations 11.10 Direct Products and Integrals If a molecule has an intrinsic symmetry, normal modes and molecular orbitals can be generated using the appropriate irreducible representations for that group Since the irreducible representations are mutually orthogonal, molecular orbitals generated from the symmetry operations will also be orthogonal Integrals of products of such orthogonal orbitals must be zero For this reason, it becomes possible to evaluate integrals involving products of functions by examining the irreducible representations of these functions The role of group theory for the evaluation of integrals is illustrated with the simple, one-dimensional function f(x) = x The function is defined on the interval ~ x ~ and must be defined on the remaining interval - ~ x ~ www.elsolucionario.net Section 11.10 • Direct Products and Integrals 641 using symmetry operations The fuH function can be defined with a C group; the actual function will depend on the irreducible representation chosen The two possibilities are shown in Figure 11.40 The A representation generates an even function while the B representation gene rates an odd function Each of these functions can be integrated over the fuH interval The even A function gives a nonzero integral while the odd B function gives a total integral of zero since the areas for the two halves of the function are equal and opposite The nonzero result is expected for the even function However, there is now an additional correlation between a nonzero integral and the A (symmetric) representation of the function The correlation persists for more complicated functions For example, the function f(x, y) = xy (11.1 0.1) can be defined for values of x and y such that 0:::; x, y:::; (11.10.2 ) in the first (positive) xy quadrant The function lies above the xy plane The symmetry operations C4 , C , and Cl can then extend this function to the remaining four x-y quadrants The choice of irreducible representation selected for the extension is now crucial If the A (totaHy symmetric) representation is chosen, the function will be identical in each quadrant and the total integral is just four times the integral determined for the first quadrant However, if the B irreducible representation is selected, the function will give a positive integral in the first and third quadrants but a negative integral in the second and fourth quadrants The total integral will be zero A total integral of zero will also be observed if the E representation is used to generate the function in aH four quadrants These simple examples illustrate a general property of functions generated using the symmetry operations and irreducible representations The integrals will be zero unless the integrand is generated using the totally symmetric irreducible representation, i.e., Al' If the function or product of functions under the integral has a finite projection on the Al irreducible representation, then the ihtegral will be nonzero o Figure 11.40 The generation of even and odd functions by application of the (a) A and (b) B irreducible representation characters to the function f(x) = x, 0",; x"'; www.elsolucionario.net 642 Chapter 11 • Introduction to Group Theory The utility of the correlation between irreducible representations and the values of integrals becomes significant when the integral contains products of functions which are genera ted using the irreducible representations for a group This situation arises frequently in studies of transitions between states in quantum mechanics where such transitions are dicta ted by matrix elements